The aim of this paper is to investigate the scheduling problem of periodic tasks in a non-DVS multiprocessor platform where each processor has multiple sleep states. We propose a task scheduling algorithm which can be divided into the offline phase and the online phase. In the offline phase, a task-to-processor assignment algorithm is utilized to derive a feasible task assignment in which the total utilization of the tasks assigned to each processor is not more than 1.0. Then, for the online phase, we propose the Break-Even Time Based Power Management (BETPM) strategy to switch a processor to a proper sleep state depending on the length of the idle time interval. A series of experiments was conducted to study the performance of the proposed algorithm. The experimental results demonstrate that the performance of the proposed algorithm is better than the compared algorithms.